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Question
f\left(x,y\right)=\sin\left(xy\right)
Function
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\text{Find the first partial derivative with respect to }x
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\text{Find the first partial derivative with respect to }y
f_{x}=y\cos\left(xy\right)
Simplify
f\left(x,y\right)=\sin\left(xy\right)
\text{Find the first partial derivative by treating the variable }y\text{ as a constant and differentiating with respect to }x
f_{x}=\frac{\partial}{\partial x}\left(\sin\left(xy\right)\right)
\text{Use the chain rule }\frac{\partial}{\partial x}(f(g))=\frac{\partial}{\partial g}(f(g))\times \frac{\partial}{\partial x}(g)\text{ where the }g\text{=}xy\text{, to find the derivative}
f_{x}=\frac{\partial}{\partial g}\left(\sin\left(g\right)\right)\times \frac{\partial}{\partial x}\left(xy\right)
\text{Use }\frac{\partial}{\partial x}(\sin x)=\cos x\text{ to find derivative}
f_{x}=\cos\left(g\right)\times \frac{\partial}{\partial x}\left(xy\right)
Evaluate
More Steps
Evaluate
\frac{\partial}{\partial x}\left(xy\right)
\text{Use differentiation rule }\frac{\partial}{\partial x}\left(cf\left(x\right)\right)=c\times\frac{\partial}{\partial x}(f(x))
y\times \frac{\partial}{\partial x}\left(x\right)
\text{Use }\frac{\partial}{\partial x} x^{n}=n x^{n-1}\text{ to find derivative}
y\times 1
Multiply the terms
y
f_{x}=\cos\left(g\right)\times y
Substitute back
f_{x}=\cos\left(xy\right)\times y
Solution
f_{x}=y\cos\left(xy\right)
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